Transforming singular quadratic forms

Question

Let $F(x_1,\ldots,x_{r+1}) = \sum_{1\leq i,j \leq r+1}A_{ij}x_ix_j$ be an integral quadartic form with rank $r$ such that $A_{11} = 0$ and $A_{12} \neq 0$. Show that there exists a unimodular transformation $U$ such that $F(U^t(x_1,\ldots,x_{r+1})) = \sum_{1\leq i,j \leq r}B_{ij}x_ix_j$, where $B_{ij}$ are also integers and $\sum_{1 \leq i,j \leq r}B_{ij}x_ix_j$ is non-singular. I am interested in this question because it is used on page 292 of a paper of G. L. Watson, which I am trying to read.